question_answer

The equation of parabola whose focus is (5, 3) and directrix is \[3x-4y+1=0\], is               [MP PET 2002]

A)            \[{{(4x+3y)}^{2}}-256x-142y+849=0\]

B)            \[{{(4x-3y)}^{2}}-256x-142y+849=0\]

C)            \[{{(3x+4y)}^{2}}-142x-256y+849=0\]

D)            \[{{(3x-4y)}^{2}}-256x-142y+849=0\]

Correct Answer: A

Solution :

           \[P{{M}^{2}}=P{{S}^{2}}\] \[\Rightarrow \] \[{{(x-5)}^{2}}+{{(y-3)}^{2}}={{\left( \frac{3x-4y+1}{\sqrt{9+16}} \right)}^{2}}\] Þ \[25({{x}^{2}}+25-10x+{{y}^{2}}+9-6x)\] \[=9{{x}^{2}}+16{{y}^{2}}+1-12xy+6x-8y-12xy\]            Þ \[16{{x}^{2}}+9{{y}^{2}}-256x-142y+24xy+849=0\]            \[\Rightarrow \]\[{{(4x+3y)}^{2}}-256x-142y+849=0.\]